Group #10709944 — subgroups

Subgroups — #10709944

Order 768.

296 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69065	C₂	1	2	yes	maximal, complemented
2	#69061	C₂	1	2	yes	maximal
3	#68065	C₂	1	1	yes	maximal, characteristic, complemented
4	#68058	C₂	1	1	yes	maximal, characteristic
5	C₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	#12526	—	3	1	no	maximal, Sylow
7	C₂×#800	—	2	4	no	—
8	#4699	C₂²	1	2	yes	—
9	#4695	C₂²	1	1	yes	characteristic
10	#4694	—	2	1	no	—
11	#4693	—	2	1	no	—
12	#4682	—	2	1	no	—
13	#4682	—	2	1	no	—
14	#4682	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₁₆	C₂²	1	2	yes	—
16	C₃×C₈²	C₂²	1	1	yes	characteristic
17	#4503	—	2	4	no	—
18	#2248	—	3	2	no	—
19	#2241	—	3	2	no	—
20	#1713	—	3	1	no	—
21	#1706	—	3	1	no	—
22	#1312	D₃	1	1	yes	characteristic, complemented
23	C₂×Dic₁₂	—	2	2	no	—
24	C₂×Dic₁₂	—	2	2	no	—
25	C₂×D₂₄	—	2	2	no	—
26	C₂×#204	—	4	2	no	—
27	#874	—	2	1	no	—
28	#869	—	2	2	no	—
29	#869	—	2	1	no	—
30	C₂×C₃×C₁₆	D₄	1	4	yes	—
31	C₃×C₄×C₈	D₄	1	1	yes	characteristic
32	C₃×C₄×C₈	D₄	1	1	yes	characteristic
33	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
34	#821	—	4	2	no	—
35	#818	—	2	2	no	—
36	#818	—	4	2	no	—
37	#818	—	4	2	no	—
38	#800	—	4	8	no	—
39	C₂×SD₃₂	—	6	4	no	—
40	#500	—	3	1	no	—
41	#500	—	6	1	no	—
42	#500	—	6	1	no	—
43	#494	—	3	2	no	—
44	#493	—	3	1	no	—
45	#492	—	6	1	no	—
46	#486	—	6	1	no	—
47	#358	—	6	4	no	—
48	C₄×C₁₆	C₂×D₃	1	2	yes	—
49	C₈²	C₂×D₃	1	1	yes	characteristic
50	C₂×D₁₂	—	4	1	no	—
51	C₂×D₁₂	—	4	1	no	—
52	C₂×D₁₂	—	4	1	no	—
53	C₂×Dic₆	—	4	2	no	—
54	C₂×Dic₆	—	4	1	no	—
55	C₂×C₃×C₈	D₈	1	2	yes	—
56	C₂×C₃×C₈	D₈	1	2	yes	—
57	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
58	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₄⋊Dic₃	—	4	1	no	—
60	C₄⋊Dic₃	—	4	2	no	—
61	C₄⋊Dic₃	—	4	2	no	—
62	C₄⋊Dic₃	—	4	1	no	—
63	Dic₁₂	—	4	4	no	—
64	Dic₁₂	—	4	4	no	—
65	D₂₄	—	4	4	no	—
66	#204	—	8	4	no	—
67	C₃×C₁₆	—	2	4	no	—
68	C₂×Q₁₆	—	6	2	no	—
69	C₂×Q₁₆	—	6	2	no	—
70	C₂×SD₁₆	—	12	2	no	—
71	C₂×D₈	—	6	2	no	—
72	C₄⋊Q₈	—	6	1	no	—
73	C₄⋊Q₈	—	6	2	no	—
74	#127	—	6	1	no	—
75	SD₃₂	—	12	8	no	—
76	C₂×C₁₆	D₁₂	1	4	yes	—
77	#107	—	6	2	no	—
78	#107	—	12	2	no	—
79	#107	—	12	2	no	—
80	#102	—	12	2	no	—
81	C₄×C₈	D₁₂	1	1	yes	characteristic
82	C₄×C₈	D₁₂	1	1	yes	characteristic
83	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
84	C₂²×D₃	—	8	1	no	—
85	C₂×C₃×C₄	#127	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
88	C₂×Dic₃	—	8	2	no	—
89	C₂×Dic₃	—	8	1	no	—
90	D₁₂	—	8	2	no	—
91	D₁₂	—	8	2	no	—
92	D₁₂	—	8	2	no	—
93	Dic₆	—	8	2	no	—
94	Dic₆	—	8	2	no	—
95	Dic₆	—	8	2	no	—
96	C₃×C₈	—	2	2	no	—
97	C₃×C₈	SD₃₂	1	4	yes	complemented
98	C₃×C₈	AGL(1,ℤ/8)	1	4	yes	—
99	C₂×Q₈	—	12	2	no	—
100	C₂×Q₈	—	12	1	no	—
101	C₂×D₄	—	12	1	no	—
102	C₂×D₄	—	12	1	no	—
103	C₂×D₄	—	12	1	no	—
104	Q₁₆	—	12	4	no	—
105	Q₁₆	—	12	4	no	—
106	SD₁₆	—	24	4	no	—
107	D₈	—	12	4	no	—
108	C₂×C₈	D₂₄	1	2	yes	—
109	C₂×C₈	D₂₄	1	2	yes	—
110	C₂×C₈	C₂×D₁₂	1	2	yes	—
111	C₄⋊C₄	—	12	1	no	—
112	C₄⋊C₄	—	12	1	no	—
113	C₄⋊C₄	—	12	2	no	—
114	C₄⋊C₄	—	12	2	no	—
115	C₄²	C₂×D₁₂	1	1	yes	characteristic
116	C₁₆	—	2	4	no	—
117	C₂²×C₃	#493	1	1	yes	characteristic
118	C₂×D₃	—	16	1	no	—
119	C₂×D₃	—	16	1	no	—
120	C₂×D₃	—	16	1	no	—
121	C₃×C₄	#496	1	2	yes	—
122	C₃×C₄	C₂×SD₃₂	1	2	yes	—
123	C₃×C₄	#510	1	2	yes	—
124	Dic₃	—	16	2	no	—
125	Dic₃	—	16	1	no	—
126	C₂³	—	24	1	no	—
127	Q₈	—	24	2	no	—
128	Q₈	—	24	2	no	—
129	Q₈	—	24	2	no	—
130	D₄	—	24	2	no	—
131	D₄	—	24	2	no	—
132	D₄	—	24	2	no	—
133	C₂×C₄	—	24	1	no	—
134	C₂×C₄	—	24	2	no	—
135	C₂×C₄	#874	1	1	yes	characteristic
136	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
137	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	#800	1	4	yes	complemented
140	C₈	#908	1	4	yes	—
141	C₂×C₃	#1721	1	1	yes	characteristic
142	C₂×C₃	#2248	1	1	yes	characteristic
143	C₂×C₃	#2251	1	1	yes	characteristic
144	D₃	—	32	1	no	—
145	C₂²	—	48	1	no	—
146	C₂²	—	48	1	no	—
147	C₂²	—	48	1	no	—
148	C₂²	#4695	1	1	yes	characteristic
149	C₄	—	48	1	no	—
150	C₄	—	48	2	no	—
151	C₄	#4712	1	2	yes	—
152	C₄	C₂×#800	1	2	yes	—
153	C₄	#4909	1	2	yes	—
154	C₃	#12526	1	1	yes	characteristic, complemented, Sylow
155	C₂	—	96	1	no	—
156	C₂	#68094	1	1	yes	characteristic
157	C₂	#69061	1	1	yes	characteristic
158	C₂	#69084	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#69065

C₂

yes

maximal, complemented

#69061

C₂

yes

maximal

#68065

C₂

yes

maximal, characteristic, complemented

#68058

C₂

yes

maximal, characteristic

C₃×#1312

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

#4695

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₃×C₄×C₁₆

C₂²

yes

—

C₃×C₈²

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂×Dic₁₂

—

C₂×Dic₁₂

—

C₂×D₂₄

—

—

—

—

—

C₂×C₃×C₁₆

D₄

yes

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

C₂³

yes

—

—

—

—

—

C₂×SD₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₃

yes

—

C₈²

C₂×D₃

yes

characteristic

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

Dic₁₂

—

D₂₄

—

#204

—

C₃×C₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

D₁₂

yes

—

—

—

—

—

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

C₂²×D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

SD₃₂

yes

complemented

C₃×C₈

AGL(1,ℤ/8)

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

Q₁₆

—

105

Q₁₆

—

106

SD₁₆

—

107

D₈

—

108

C₂×C₈

D₂₄

yes

—

109

C₂×C₈

D₂₄

yes

—

110

C₂×C₈

C₂×D₁₂

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

C₂×D₁₂

yes

characteristic

116

C₁₆

—

117

C₂²×C₃

#493

yes

characteristic

118

C₂×D₃

—

119

C₂×D₃

—

120

C₂×D₃

—

121

C₃×C₄

#496

yes

—

122

C₃×C₄

C₂×SD₃₂

yes

—

123

C₃×C₄

#510

yes

—

124

Dic₃

—

125

Dic₃

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

D₄

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

#874

yes

characteristic

136

C₂×C₄

C₂×D₂₄

yes

characteristic

137

C₂×C₄

C₂×D₂₄

yes

characteristic

138

C₈

—

139

C₈

#800

yes

complemented

140

C₈

#908

yes

—

141

C₂×C₃

#1721

yes

characteristic

142

C₂×C₃

#2248

yes

characteristic

143

C₂×C₃

#2251

yes

characteristic

144

D₃

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#4695

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

#4712

yes

—

152

C₄

C₂×#800

yes

—

153

C₄

#4909

yes

—

154

C₃

#12526

yes

characteristic, complemented, Sylow

155

C₂

—

156

C₂

#68094

yes

characteristic

157

C₂

#69061

yes

characteristic

158

C₂

#69084

yes

characteristic

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1312	C₂
socle	C₂²×C₃	#493
prime radical	#4695	C₂²
hypercenter	#1312	D₃
hypocenter	C₃	#12526
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#4695

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂²

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #69065	C₂	semidirect
3. #68065	C₂	semidirect
5. C₃×#1312	C₂	semidirect
22. #1312	D₃	semidirect
97. C₃×C₈	SD₃₂	semidirect
139. C₈	#800	semidirect
154. C₃	#12526	semidirect

H ≅ G/N

type

1. #69065

C₂

semidirect

3. #68065

C₂

semidirect

5. C₃×#1312

C₂

semidirect

22. #1312

D₃

semidirect

97. C₃×C₈

SD₃₂

semidirect

139. C₈

#800

semidirect

154. C₃

#12526

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#69065	2
#69061	2
#68065	2
#68058	2
C₃×#1312	2
#12526	3

subgroup

index

Subgroups — #10709944

Conjugacy classes of subgroups

Complements

Maximal subgroups